If for some positive integer $mathrm{n}$, the coefficients of three consecutive terms in the binomial

Question:

If for some positive integer $\mathrm{n}$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$, then the largest coefficient in this expansion is :-

  1. 792

  2. 252

  3. 462

  4. 330


Correct Option: , 3

Solution:

Let $\mathrm{n}+5=\mathrm{N}$

$\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}=5: 10: 14$

$\Rightarrow \frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}}}}{\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}}=\frac{\mathrm{N}+1-\mathrm{r}}{\mathrm{r}}=2$

$\frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}}{\mathrm{~N}_{\mathrm{C}_{\mathrm{t}}}}=\frac{\mathrm{N}-\mathrm{r}}{\mathrm{r}+1}=\frac{7}{5}$

$\Rightarrow \quad r=4, N=11$

$\Rightarrow(1+x)^{11}$

Largest coefficient $={ }^{11} \mathrm{C}_{6}=462$

 

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